A231132 -id:A231132 - cp's OEIS frontend

A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

1, 0, 4, 3, 4, 0, 2, 9, 1, 7, 5, 7, 4, 2, 8, 8, 7, 3, 3, 2, 5, 5, 2, 8, 9, 6, 4, 6, 6, 7, 1, 6, 7, 6, 0, 3, 0, 5, 4, 8, 4, 7, 0, 8, 6, 6, 0, 4, 6, 8, 8, 2, 5, 6, 1, 0, 4, 4, 5, 7, 0, 4, 7, 9, 7, 6, 9, 5, 8, 5, 0, 6, 2, 5, 5, 2, 5, 2, 4, 8, 4, 3, 2, 7, 6, 1, 5, 1, 0, 7, 2, 0, 7, 9, 8, 4, 1, 4, 3, 5, 6, 2, 1, 4, 6

Offset: 1

Vaclav Kotesovec, Mar 09 2019

Sum_{k>=2} (-1)^k*Zeta(k)/k = A001620 (see MathWorld, formula 122).

			1.043402917574288733255289646671676030548470866046882561044570479769585...

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A231132, A306760, A306765, A306774, A306778, A307106.

evalf(Sum((-1)^j*Zeta(j)^2/j, j=2..infinity), 100);

NSum[(-1)^k*Zeta[k]^2/k, {k, 2, Infinity}, WorkingPrecision -> 200, NSumTerms -> 100000]

sumalt(k=2, (-1)^k*zeta(k)^2/k) \\ Michel Marcus, Mar 09 2019

Equals log(A306765) + A001620^2.

A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

Original entry on oeis.org

2, 0, 3, 4, 4, 4, 8, 9, 4, 5, 4, 8, 7, 6, 1, 6, 4, 7, 7, 9, 8, 0, 3, 5, 5, 5, 3, 1, 8, 8, 6, 9, 0, 2, 6, 3, 5, 5, 9, 7, 9, 4, 3, 9, 8, 6, 3, 7, 0, 2, 3, 7, 6, 2, 6, 0, 0, 0, 5, 2, 8, 4, 1, 6, 5, 6, 5, 0, 0, 7, 8, 2, 7, 7, 5, 7, 1, 1, 3, 2, 4, 4, 5, 0, 2, 6, 5, 0, 4, 0, 6, 1, 3, 5, 0, 7, 5, 0, 2, 9, 1, 2, 7, 1, 4

Offset: 1

Vaclav Kotesovec, Mar 08 2019

			2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

Vaclav Kotesovec, Table of n, a(n) for n = 1..297

Cf. A001620, A231132, A303670, A306760, A306769.

evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);

slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]

exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.
Equals exp(-gamma^2 + A306769).
Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(k).

A365959 Decimal expansion of Sum_{k>=2} zeta(k)/k^2.

Original entry on oeis.org

8, 3, 5, 9, 9, 8, 3, 3, 2, 7, 0, 0, 9, 6, 4, 3, 2, 2, 9, 7, 0, 9, 1, 1, 1, 9, 8, 6, 9, 6, 0, 2, 9, 0, 9, 6, 4, 2, 7, 0, 4, 2, 1, 6, 8, 0, 9, 3, 2, 3, 3, 2, 4, 8, 3, 2, 9, 5, 5, 6, 3, 4, 9, 2, 5, 7, 7, 0, 1, 8, 9, 5, 2, 5, 3, 1, 7, 3, 8, 8, 4, 0, 0, 2, 1, 9, 2, 2, 7, 6, 8, 8, 7, 7, 6, 6, 2, 2, 3, 2, 5, 7, 7, 8, 5, 5

Offset: 0

Vaclav Kotesovec, Sep 23 2023

			0.835998332700964322970911198696029096427042168093233248329556349257701895253...

Mathematics Stack Exchange, On a log-gamma definite integral, 2020.
Michael I. Shamos, A catalog of the real numbers, 2011, p. 596.

Cf. A231132, A256921.

PARI
```
sumpos(k=2,zeta(k)/k^2)
```

Equals Sum_{k>=1} (polylog(2, 1/k) - 1/k).
From Velin Yanev, Jul 30 2025: (Start)
Equals Integral_{x=0..1} log(Gamma(1 - x))/x dx - A001620. [Proved by Paul Enta, 2020]
Conjecture: Equals 2 - A001620 - Pi^2/12 + Integral_{x=0..oo} (2*x*arccot(x) - log(1/x^2 + 1))*log(1 - exp(-2*Pi*x))/(2*Pi*(x^2 + 1)) dx. (End)

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A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

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A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

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A365959 Decimal expansion of Sum_{k>=2} zeta(k)/k^2.

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